THE CRITICAL VARIATIONAL SETTING FOR STOCHASTIC EVOLUTION EQUATIONS

. In this paper we introduce the critical variational setting for parabolic stochastic evolution equations of quasi-or semi-linear type. Our results improve many of the abstract results in the classical variational setting. In particular, we are able to replace the usual weak or local monotonicity condition by a more ﬂexible local Lipschitz condition. Moreover, the usual growth conditions on the multiplicative noise are weakened considerably. Our new setting provides general conditions under which local and global existence and uniqueness hold. Moreover, we prove continuous dependence on the initial data. We show that many classical SPDEs, which could not be covered by the classical variational setting, do ﬁt in the critical variational setting. In particular, this is the case for the Cahn-Hilliard equations, tamed Navier-Stokes equations, and Allen-Cahn equation.


Introduction
The variational approach to stochastic evolution equations goes back to the work of [8,28,35].Detailed information on the topic can also be found in the monographs [31,41].Further progress was made in the papers [11], where the Lévy case was considered, and in [30], where the local monotonicity conditions are weakened.Recently in the papers [20,34] a new type of coercivity condition has been found which gives sufficient conditions for L p (Ω)-estimates as well.
As usual we assume that (V, H, V * ) is a triple of spaces such that V → H → V * , where we use the identification of the Hilbert space H and its dual, and where V * is the dual with respect to the inner product of H.In the classical framework V can be a Banach space, but since we will be dealing with non-degenerate quasi-linear equations only, it will be natural to assume V is a Hilbert space as well.The stochastic evolution equation we consider can be written in the abstract form: (1.1) du(t) + A(t, u(t))dt = B(t, u(t))dW (t), u(0) = u 0 .
Here W is a U -cylindrical Brownian motion and U is a separable Hilbert space.One of the advantages of the variational approach is that it gives global well-posedness for a large class of stochastic evolution equations.Other approaches typically only give local well-posedness and sometimes (after a lot of hard work) global well-posedness under additional conditions.
1.1.The classical variational setting.In the variational setting there are several conditions on the nonlinearities (A, B).The most important one is: which allows to use Itô's formula to deduce a priori bounds for the solution.Moreover, in the proof of these bounds, but also at other places, one needs some boundedness properties of (A, B): In many cases the leading order part of A is linear or quasi-linear and uniformly elliptic, and in that case, the coercivity condition usually forces one to take α = 2, in which case B needs to be of linear growth.Moreover, the growth condition on A becomes . From an evolution equation perspective it seems more natural to have a symmetric growth condition in terms of intermediate spaces between H and V for the non-leading part of A, which is exactly what we assume later on.
In many examples a problematic condition is the local monotonicity condition: One of the difficulties with this is that the "one-sided Lipschitz constant" on right-hand side can only grow as v β H (or equivalently u β H ), but not in u β H and v β H at the same time.Typically it fails whenever, spatial derivatives of u appear in a nonlinear way.Examples of nonlinearities for where local monotonicity fails are • the Cahn-Hilliard nonlinearity ∆f (u), where f (y) = ∂ y [ 1  4 (1 − y 2 ) 2 ], • the Navier-Stokes nonlinearity (u • ∇)u in three dimensions, • the Allen-Cahn nonlinearity u − u 3 in two or three dimensions, • systems of SPDEs with nonlinearities such as ±u 1 u 2 .For the Navier-Stokes and Allen-Cahn equations the strong setting V = H 2 and H = H 1 is required, for which local monotonicity does not hold.Each of the above cases contains a product term, which does not satisfy local monotonicity.
The local monotonicity conditions is further weakened in [31,Section 5.2], where also some of the above examples are considered.However, therein global well-posedness is only obtained for equations with additive noise.In what follows, we introduce a new setting in which we can remove such restrictions.
1.2.The critical variational setting.In this paper we introduce a new variational setting which we call the critical variational setting.We show that under mild structural restrictions on (A, B), one can replace the local monotonicity condition by a local Lipschitz condition, which does hold for the above mentioned examples.The Lipschitz constants can have arbitrary dependence on u H and v H , and moreover they can grow polynomially in u V β and v V β , where V β = [V * , V ] β is the complex interpolation space and β < 1.A restriction on the nonlinearities (A, B) is that they are of quasi-linear type: where v ∈ V .The parts (A 0 , B 0 ) and (F, G) are the quasi-linear and semi-linear parts of (A, B), respectively.Of course many of the classical SPDEs fit into this setting, and actually in many important cases A 0 and B 0 are linear differential operators, which requires α = 2 in the coercivity condition (1.2).
To give an idea what to expect we give a special case of the local Lipschitz condition on F and G we will be assuming: for all n ≥ 1 there exists a C n s.t. for all u, v ∈ V with u H ≤ n and v H ≤ n one has Here β ∈ (0, 1) and ρ ≥ 0 satisfy 2β ≤ 1 + 1 ρ+1 , together with an associated growth condition involving the same parameters β and ρ.In case 2β = 1 + 1 ρ+1 the nonlinearity is called critical.It is central in the theory that this case is included as well, as often it has the right scaling properties for SPDEs.In examples, the Sobolev embedding usually dictates which β one can use.Then the corresponding ρ, gives the growth of the Lipschitz constant which one can allow.
Another feature of our theory is that we can also weaken the linear growth condition on the B-term in front of the noise term (i.e.(1.4) in case α = 2).In principle we can have arbitrary growth, but in order to ensure global well-posedness, the coercivity condition usually leads to some natural restrictions on the growth order of B.
Finally, we mention that the classical variational setting also covers several degenerate nonlinear operators A such as the p-Laplace operator and the porous media operator (see [31]).At the moment we cannot treat these cases since [3,4] requires a quasi-linear structure.It would be interesting to see whether the latter can be adjusted to include these cases as well.This could lead to a complete extension of the variational setting where the local monotonicity condition is replaced by local Lipschitz conditions.After the first version of the current paper was uploaded to the arXiv, the manuscript [39] appeared on the arXiv.In the latter it was shown that the classical variational framework can also be extended to a fully nonlinear framework under very general local monotonicity conditions in case the embedding V → H is compact, by applying Yamada-Watanabe theory.The approaches are completely different, but there is some overlap in the potential applications.
1.3.Our results.In the paper we prove well-posedness for (1.1) in the critical variational setting, i.e. assuming (1.5), a coercivity condition, and a local Lipschitz condition.The main results for the abstract stochastic evolution equation (1.1) are • Global well-posedness (Theorems 3.4 and 3.5); • Continuous dependency on the initial data (Theorem 3.8).The abstract results are applied to the following examples: • Stochastic Cahn-Hilliard equation (Section 5.1); • Stochastic Tamed Navier-Stokes equations (Section 5.2); • Generalized Burgers equations (Section 5.3); • Allen-Cahn equation (Section 5.4); • A quasi-linear equations of second order (Section 5.5); • Stochastic Swift-Hohenberg equations (Section 5.6).Some of the applications have not been considered before, because the classical variational framework was not applicable.In all cases the proofs are new and lead to new insights.The new variational setting avoids local monotonicity and boundedness, and thus leads to less restrictions.Many more applications could be considered, but to keep the paper at a reasonable length, we have to limit the number of applications.We kindly invite the reader to see what the new theory brings for his/her favorite SPDEs.
Our method of proof differs from the classical variational setting which is based on Galerkin approximation.Local well-posedness is an immediate consequence of our paper [3], which is the stochastic analogue of the theory of critical evolution equations recently obtained in [29,37,38].
To obtain global well-posedness, we use the coercivity condition to derive an a priori estimate via Itô's formula.Using a variation of our blow-up criteria, recently obtained in [4], we deduce global existence.Continuous dependence on the initial data turns out to be a delicate issue, and is proved via maximal regularity techniques combined with a local Gronwall lemma.
1.4.Embedding into L p (L q )-theory.In follow-up papers we will also present other examples which require L p (L q )-theory (L p in time and L q in space).Choosing p and q large, Sobolev embedding results become better, and more nonlinearities can be included.In many cases the L 2 (L 2 )-setting can be transferred to a L p (L q )-setting using the new bootstrap methods of [4,Section 6].Often this also leads to global well-posedness in an L p (L q )-setting.
An example where the latter program is worked out in full detail can be found in [2].Here we proved higher order regularity of local solutions to the stochastic Navier-Stokes equations on the d-dimensional torus.In the special case of d = 2, these results hold globally and are proved using such a transference and bootstrapping argument, by going from an L 2 (L 2 )-setting to an L p (L q )-setting.
There is a price to pay in L p (L q )-theory.It is usually much more involved, and moreover, the classical coercivity conditions are often not enough or not even known.Exceptions where coercivity conditions in L p (L q ) are well-understood are: • second order equations on R d [27,36]; • second order equations on the torus T d [1]; • weighted spaces on domains [23,24].It is of major importance to further extend the latter linear theory as it is the key in the stochastic maximal regularity (SMR) approach to SPDEs obtained in [3,4].In future papers we expect that the variational setting considered in the current paper will play a crucial role in establishing global L p (L q )-theory, which in turn also leads to higher order regularity of the solution.
Notation.For a metric space (M, d), B(M ) denotes the Borel σ-algebra.
For a measure space (S, A, µ), L 0 (S; X) denotes the space of strongly measurable functions from S into a normed space X.Here and below we identify a.e.equal functions.For details on strong measurability we refer to [21].For p ∈ (0, ∞) let L p (S; X) denote the subset of L 0 (S; X) for which Finally, let L ∞ (S; X) denote the space of functions for which In case A 0 is a sub-σ-algebra of A, L p A0 (S; X) denotes the subspace of strongly A 0 -measurable functions in L p (S; X). Let We write L p loc (I; X) (resp.C loc (I)) for the space of functions f : For a Hilbert spaces U and H, let L(U, H) denote the bounded linear operators from U into H, and L 2 (U, H) the Hilbert-Schmidt operators from U into H.
Given Hilbert spaces β denotes complex interpolation at β ∈ (0, 1), and However, in case β ∈ {0, 1/2, 1} we also write By the notation A p B we mean that there is a constant C depending only on p such that A ≤ CB.
In the variational setting for evolution equations one starts with two real Hilbert spaces (V, (•, •) V ) and (H, (•, •) H ) such that V → H, where the embedding is dense and continuous.Identifying H with its dual we can define an imbedding j : H → V * by setting v, jh = (v, h) H for v ∈ V and h ∈ H.In this way we may view H as a subspace of V * , and one can show that the embedding H → V * is dense and continuous.Moreover, one can check that ( A converse to the above construction can also be done.Given Hilbert spaces V 0 and V 1 such that V 1 → V 0 , where the embedding is dense and continuous, define another Hilbert space H by setting In case of second order PDEs with Dirichlet boundary conditions (cf.Subsection 5.3), the weak setting is obtained by letting In case the domain is C 1 the celebrated result of Seeley [42] yields Note that the choice V = H 1 and 2.2 can also be extended to the case of Neumann boundary conditions.In the weak setting this leads to H = H 1 (O) and V = L 2 (O).In the strong setting this leads to V = {u ∈ H 2 (O) : ∂ n u| ∂O = 0} and H = H 1 (O).Extensions to higher order operators can be considered as well.

Stochastic calculus.
For details on stochastic integration in Hilbert spaces we refer to [13,31,41] and for details on measurability and stopping times to [22].For completeness we introduce the notation we will use.
Let H be a Hilbert space.Let (Ω, F, P) be a probability space with filtration (F t ) t≥0 .An H-valued random variable ξ is a strongly measurable mapping ξ : Ω → H.
For a real separable Hilbert space U , let (W (t)) t≥0 be a U -cylindrical Brownian motion with respect to (F t ) t≥0 , i.e.W ∈ L(L 2 (R + ; U ), L 2 (Ω)) is such that for all f, g ∈ L 2 (R + ; U ) one has (a) W f has a normal distribution with mean zero and If Φ : R + ×Ω → L 2 (U, H) is strongly progressively measurable and Φ ∈ L 2 loc (R + ; L 2 (U, H)) a.s., then one can define the stochastic integral process • 0 ΦdW , which has a version with continuous paths a.s., and for each interval J ⊆ [0, ∞) the following two-sided estimate holds where p ∈ (0, ∞) and C p > 0 only depends on p.The latter will be referred to as the Burkholder-Davis-Gundy inequality.

Main results
As in Section 2.1 let V 0 , V 1 and H be Hilbert spaces such that . By reiteration for the real or complex interpolation method (see [9, p. 50 and 101]) one also has that The following short-hand notation will be used frequently: where U is the Hilbert space for the cylindrical Brownian motion W .However, in case β ∈ {0, 1/2, 1} we prefer to write The following interpolation estimate will be frequently used As explained in Section 2.1 we will also write V = V 1 and V * = V 0 , where the duality relation is given by v and extended to a mapping •, • : V × V * → R by continuity.
3.1.Setting.Let W be a U -cylindrical Brownian motion (see e.g.[3,Definition 2.11]).Consider the following quasi-linear stochastic evolution equation: The following conditions will ensure local well-posedness.
where t ∈ [0, T ], u ∈ V , and v ∈ H satisfies v H ≤ n. (3) Let ρ j ≥ 0 and β j ∈ (1/2, 1) be such that where m F , m G ∈ N, and suppose that ∀n ≥ 1 ∀T > 0 there exists a constant C n,T such that a.s. ), Here (2) means that the linear part (A 0 (•, v), B 0 (•, v)) is a usual coercive pair with locally uniform estimates for v H ≤ n.Moreover, the quasi-linearities x → A 0 (•, x) and x → B 0 (•, x) are locally Lipschitz.In the semi-linear case, i.e. when A 0 and B 0 do not depend on x, the conditions on A 0 and B 0 in (3) become trivial.
The nonlinearities F and G satisfy a local Lipschitz and growth estimate, which contains several tuning parameters β j and ρ j such that (3.4) holds, or equivalently (2β j − 1)(ρ j + 1) ≤ 1. Usually m F = m G = 1, so that the sums on the left-hand side disappear.The case of equality in (3.4) for some j leads to so-called criticality of the corresponding nonlinearity.The growth in the Hnorm can be arbitrary large and this is contained in the constant C n,T .These types of conditions have been introduced for deterministic evolution equation in [29,37,38], and in the stochastic framework in [3,4] in a Banach space setting.Definition 3.2 (Solution).Let Assumption 3.1 be satisfied and let σ be a stopping time with values in [0, ∞].Let u : [0, σ) × Ω → V be a strongly progressively measurable process.
and the following identity holds a.s. and for all t ∈ [0, σ), In this case we write u instead of (u, σ).
Maximal local solutions are always unique in the class of local solutions.This seems to be a stronger requirement compared to [3,Definition 4.3].However, due to the coercivity condition on the leading operator (A 0 , B 0 ), i.e.Assumption 3.1(2), stochastic maximal L 2 -regularity holds by Lemma 4.1 below.Thus [4,Remark 5.6] shows that Definition 3.2 coincide with the one in [3,Definition 4.3].
Under Assumption 3.1 the following local well-posedness result holds, which will be proved in Section 4.

3.2.
Global well-posedness.Under a coercivity condition we obtain two analogues of the wellposedness result in the classical variational framework.Theorem 3.4 (Global well-posedness I).Suppose that Assumptions 3.1 holds and for all T > 0, there exist η, θ, M > 0 and a progressively measurable φ ∈ L 2 ((0, T ) × Ω) and for any v ∈ V and Then for every u 0 ∈ L 0 F0 (Ω; H), there exists a unique global solution u of (3.3) such that a.s.
Moreover, for each T > 0 there is a constant C T > 0 independent of u 0 such that In the above result we do not assume any growth conditions on A and B besides the local conditions in Assumptions 3.1.The additional η > 0 in the coercivity condition (3.8) can be arbitrary small and in most examples it does not create additional restrictions.Setting η = 0 in the coercivity condition (3.8), it reduces to the standard one in the variational approach to stochastic evolution equations (see [31,Section 4.1]).From a theoretical perspective it is interesting to note that we can also allow η = 0. Theorem 3.5 (Global well-posedness II).Suppose that Assumption 3.1 holds, and for all T > 0, there exist θ, C, M > 0 and a progressively measurable φ ∈ L 2 ((0, T ) × Ω) and for any v ∈ V and Then for every u 0 ∈ L 0 F0 (Ω; H), there exists a unique global solution u of (3.3) such that a.s.
The latter result shows that the full statement of the classical variational setting can be obtained in our setting (see Assumption 3.1) without assuming the growth condition (1.3) and without the local monotonicity condition.
Remark 3.6 (Variants of the coercivity condition).The coercivity condition (3.8) can be replaced by the following weaker estimate To see this, one can easily adapt the proof of Theorem 3.5 which will be given under the more restrictive condition (3.8).Since it does not give more flexibility in the examples we consider, we prefer to work with (3.8).A variant of condition (3.14) with η = p−2 2 with p > 2 is considered in [20], where it is used to establish bounds on higher order moments of the solution.
In case (3.8) holds with η = 0 and Remark 3.7 (Ω-localization of (u 0 , φ)).The function φ is used to take care of the possible inhomogeneities f and g (see Assumption 3.1(1)).We have only considered the case that φ ∈ L 2 ((0, T )×Ω) for all T > 0. However, by a standard stopping time argument and using the uniqueness of solutions to (3.3), one can also consider the case that for all T > 0 a.s.φ ∈ L 2 (0, T ).Moreover, the estimates of Theorems 3.4-3.5 can be used to prove tail estimates for u.For instance, (3.9) and the Chebychev inequality readily yield, for all R, γ > 0, For more details see the proof of Lemma A.1 where a similar argument is employed.
A similar argument can also be employed to obtain tail estimates in case u 0 ∈ L 0 (Ω; H).

3.3.
Continuous dependence on initial data.Now that we also have global existence of solutions, we can consider the question whether one has continuous dependence on the initial data.This indeed turns out to be the case in the setting of both of the above results.In the case where the monotonicity holds, continuous dependence is immediate from Itô's formula (see [31,Proposition 4.2.10]).The following result requires no monotonicity conditions at all, and only assumed the conditions we already imposed for global existence and uniqueness: Theorem 3.8 (Continuous dependence on initial data).Suppose that the conditions of Theorem 3.4 or 3.5 hold.Let u and u n denote the unique global solutions to (3.3) with strongly F 0measurable initial values u 0 and u 0,n , respectively.Suppose that u 0,n → u 0 in probability.Let T ∈ (0, ∞).Then If additionally, sup n≥1 u 0,n L 2 (Ω;H) < ∞ then for any q ∈ (0, 2) Remark 3.9 (Feller property).Suppose that the assumptions of Theorem 3.4 (resp.3.5) hold and let u ξ be the global solution to (3.3) with data ξ ∈ H.As usual, one defines the operator for t ≥ 0, ξ ∈ H and ϕ ∈ C b (H).Theorem 3.8 shows that P t : C b (H) → C b (H).This is usually referred as Feller property and this is the starting point for investigating existence and uniqueness of invariant measures for (3.3).However, this is not the topic of this paper.

Proofs of the main results
In several cases we need an a priori estimate for the solution to the linear equation where λ is a stopping time with values in [0, T ], u λ : Ω → H is strongly F λ -measurable, and ( A, B) are linear and satisfy the standard variational condition: A solution to (4.1) is defined in a similar way as in Definition 3.2.
The following estimate is well-known in case λ is non-random (see [31,Theorem 4.2.4] and its proof).The random case can be obtained by approximation by simple functions (see [4,Proposition 3.9]).H)) be strongly progressively measurable and suppose that there exist M, θ > 0 such that a.s.
and there is a constant C independent of (f, g, u 0 ) such that In the above F λ denotes the σ-algebra of λ-past.By [4, Proposition 3.9 and 3.12], if the stochastic maximal L 2 -regularity estimate holds on some stochastic interval [λ, T ], then it also holds on [τ, T ] for all stopping time τ ∈ [λ, T ].Moreover, the constant in the estimate can be chosen to be independent of τ .
4.2.Proof of Theorem 3.4: global well-posedness.We first prove the following global energy estimates.In the proof of Theorem 3.4 we will see that σ = ∞ in the result below.Proposition 4.3.Suppose that Assumption 3.1 holds, and for all T > 0, there exist η, θ, M > 0 and a progressively measurable φ ∈ L 2 ((0, T ) × Ω) and for any v ∈ V and t ∈ [0, T ], (Ω; H).Let (u, σ) be the maximal solution to (3.3) provided by Theorem 3.3.Then for every T > 0 there is a constant C T > 0 independent of u 0 such that where inf ∅ := τ k .Then (σ k ) k≥1 is a localizing sequence as well.Letting It is suffices to find a C > 0 independent of (k, u 0 ) such that for all t ∈ [0, T ], (4.9) Indeed, from the latter Gronwall's lemma first gives This combined with (4.9) implies It remains to let k → ∞ in the latter two estimates.The proof of (4.9) will be divided into two steps.
Step 2: Estimating the martingale part on the RHS (4.10).Set The Burkholder-Davis-Gundy inequality implies where in the last step we applied (4.11).Taking E[sup s∈[0,t] | • |] in (4.10) and using the above estimate we obtain The energy estimate of Proposition 4.3 allows us to prove Theorem 3.4 via the blow-up criteria (3.7) of Theorem 3.3.
Proof of Theorem 3.4.The proof is divided into two steps.
From the claim (4.12) and the blow-up criteria (3.7) it follows that Therefore, σ ≥ T a.s., and since T was arbitrary, we obtain σ = ∞ a.s.

4.3.
Proof of Theorem 3.5.First we obtain global estimates under the sharper coercivity estimate.In Theorem 3.5 we will see that σ = ∞ in the results below.
Proof.We begin by repeating the localization argument used in the proof of Proposition 4.3.Let (τ k ) k≥1 be a localizing sequence for (u, σ ∧ T ).Then in particular, where we set inf ∅ = τ k .Then (σ k ) k≥1 is a localizing sequence for (u, σ) as well.Letting The idea will be to eventually apply a stochastic Gronwall lemma.In order to do this let Step 1: Proof of the estimate (4.15) below.By Itô's formula (see [31,Theorem 4 where Therefore, by the coercivity condition we find that (4.15) Step 2: u ∈ L 2 (Ω; L 2 (0, τ ; V )) and the estimate (4.20) below holds.Taking expectations in (4.15) gives that for all t ∈ [0, T ], (4.16) Therefore, by the classical Gronwall inequality applied to Thus also (4.18) Applying (4.17) and (4.18) in (4.16) also yields the estimate Letting k → ∞ in (4.18) and (4.19), by Fatou's lemma we obtain that In particular (4.13) follows.
By the estimates of Proposition 4.4 we can check the blow-up criteria (3.7): Proof of Theorem 3.5.The proof is divided into two steps. Step Therefore, σ ≥ T a.s., and since T was arbitrary, we obtain σ = ∞ a.s.
Step 2: A priori bounds.The bounds (3.11) and (3.13) are immediate from (4.13) and (4.14).To prove, (3.12) note that we can replace all σ k 's by T in the proof of Proposition 4.4, and thus the required bound follows from (4.17).4.4.Proof of Theorem 3.8: continuous dependence on initial data.In this section we use the notation Z := C([0, T ]; H) ∩ L 2 (0, T ; V ).The following tail estimate is the key to the proof of the continuous dependency.Proposition 4.5.Suppose that the conditions of Theorem 3.5 hold.Let u and v denote the solution to (3.3) with initial values u 0 and v 0 in L 0 F0 (Ω; H), respectively, and where u 0 L ∞ (Ω;H) + v 0 L ∞ (Ω;H) ≤ r for some r > 0. Then for every T > 0 there exist ψ 1 , ψ 2 : [r, ∞) → (0, ∞) both independent of u 0 and v 0 such that lim R→∞ ψ 2 (R) = 0 and for all ε > 0 and R ≥ r where f w = f 1 + f 2 and g w = g 1 + g 2 are given by In order to derive an a priori estimate for w, we want to apply Lemma 4.1 to the pair (A 0 (•, u), B 0 (•, u)).
In order to check (4.2) we will use Assumption 3.1(2) and a suitable stopping time argument to ensure where we used (3.13) with λ = 1/2 in the last step.
It remains to estimate P w Z τ R 0 ≥ ε .To do so we apply the stochastic Gronwall Lemma A.1.Let 0 ≤ λ ≤ Λ ≤ τ R be stopping times.Let w be the solution to where the constant C R in (4.23) is independent of (λ, Λ) (see below Lemma 4.1).
Next we estimate the f w and g w term in terms of u, v and w by using Assumption 3.1(3).Without loss of generality (by increasing ρ j if necessary) we can assume 2β j = 1 + 1 ρj +1 for every j ∈ {1, . . ., n}.Then , we obtain where we used Young's inequality with exponents 1/(2 − 2β j ) and 1/(2β j − 1), and the fact that Adding the (A, B)-estimate, we can conclude that Combining this with (4.23) and choosing δ R > 0 small enough we find that Since u, v ∈ L 2 (0, T ; V ) a.s. and C R is independent of (λ, Λ), we can apply the stochastic Gronwall lemma A.1 with R replaced by C R R, to find a constant K R such that where we used (3.11).Thus the required estimate follows.
After these preparation we can now prove the continuous dependence result.
Proof of Theorem 3.8.It suffices to prove the result under the conditions of Theorem 3.5 as they are weaker than the conditions of Theorem 3.8.In Step 1 we deal with the uniformly bounded case, and in Step 2 we reduce to this case.Below we fix T > 0 and recall that Z = C([0, T ]; H) ∩ L 2 (0, T ; V ).
Step 1: Case of uniformly bounded initial data.Suppose that there exists a constant r > 0 such that u 0,n H + u 0 H ≤ r a.s.for all n ≥ 1.We will show that for every ε > 0, Let ε > 0. By Proposition 4.5 we obtain that for all R ≥ r Since ψ 2 (R) → 0 as R → ∞, (4.24) follows.
Step 2: General case.We will show that u n → u in Z in probability.By considering subsequences we may additionally suppose that u 0,n → u 0 in H a.s.Let δ > 0. By Egorov's theorem we can find Ω 0 ∈ F 0 and k ≥ 1 such that P(Ω 0 ) ≥ 1 − δ and u 0,n H ≤ k on Ω 0 .

Applications to stochastic PDEs
Throughout this section (w n t : t ≥ 0) n≥1 denotes a sequence of standard Brownian motions on a probability space (Ω, F, P) with respect to a filtration (F t ) t≥0 .To such sequence one can associate an 2 -cylindrical Brownian motion by setting We make the following assumptions on f and g: The standard example f (y) = y(y 2 − 1) = ∂ y [ 1  4 (1 − y 2 ) 2 ] (double well potential) satisfies the above conditions for d ∈ {1, 2}.Note that in this case the nonlinearity ∆f (u) does not satisfy the classical local monotonicity condition for stochastic evolution equations, and therefore there are difficulties in applying the classical framework to obtain well-posedness for (5.1).In the case of additive noise, well-posedness was studied in [31, Example 5.2.27 and Remark 5.2.28].Our setting applies in the setting of multiplicative noise under the same conditions on the nonlinearity ∆f (u).
s. and for every T > 0 there exists a constant C T independent of u 0 such that Finally, u depends continuously on the initial data u 0 in probability in the sense of Theorem 3.8 with Proof.We first formulate (5.1) in the form (3.3).Let H = L 2 (O) and V = H 2 N (O).By (3.1) for θ ∈ (0, 1) one has where in the last step we used the smoothness of O and standard results on complex or real interpolation.
Remark 5.3.It is also possible to add a noise term of the form: where we need the stochastic parabolicity condition: for some λ > 0 and all u ∈ V , Depending on the precise form of f , one can allow superlinear g as well.The only requirement is for some η > 0. However, for constant functions u, the right-hand side only grows quadratically.

Stochastic tamed Navier-Stokes equations.
In [40] the stochastic tamed Navier-Stokes equations with periodic boundary conditions were considered.In [10] the problem was studied on R 3 .Both these papers construct martingale solutions and prove pathwise uniqueness.Below we show that one can argue more directly using our framework.We will consider the problem on the full space, but the periodic case can be covered by the same method.Remarks about the case of Dirichlet boundary conditions can be found in Remark 5.6.
On R 3 consider the tamed Navier-Stokes equations with gradient noise: (5.6) Here In the deterministic setting the motivation to study (5.6) comes from the fact if u is a strong solution to the usual Navier-Stokes equations and u 2 L ∞ ((0,T )×R 3 ) ≤ N , then it is also a solution to (5.6) for N large enough.On the other hand, it is possible to give conditions under which (5.6) has a unique global strong solution.
Note that, as in previous works on (5.6), our stochastic parabolicity condition (5.7) is not optimal.This is due to the fact that the taming term −φ N (|u| 2 )u needs to be handled as well. Let , where for k ∈ {0, 1, 2} we set )) be the Helmholtz projection, i.e. the orthogonal projection onto H k for k ∈ {0, 1, 2}.After applying the Helmholtz projection, (5.6) can be rewritten as (3.3)where (5.10) From Theorem 3.4 we will derive the following result.
Step 1: Assumption 3.1(2) holds.Let ε > 0 be fixed.Then ) ≤ 1 and commutes with derivatives, for the B 0 -part we can write By (5.7) the first summand satisfies The second summand is of second order term and satisfies The C ε -term is again of first order: For the second order term T 2 , by (5.7), we can write For later purposes we note that this gives On the other hand, by integration by parts (and approximation) Therefore, collecting terms and choosing ε > 0 small enough we obtain that Step 2: F 1 , F 2 and G in (5.10) satisfy Assumption 3.1.For u, v ∈ V with u H ≤ n and v H ≤ n, we have ,2 , where we used Hölder's inequality with .Setting Therefore, we can set ρ 1 = 1 and thus the required condition in Assumption 3.1 since 2β 1 ≤ 1 + 1 ρ1+1 .For F 2 note that for u, v ∈ V with u H ≤ n and v H ≤ n we have where we used Hölder's inequality in (i), and Sobolev embedding in (ii).Thus F 2 satisfies the required condition in Assumption 3.1 with any β 2 ∈ ( 1 2 , 1) and ρ 2 = 0, so in particular we could take β 2 = β 1 and ρ 2 = ρ 1 as before.
For G let u, v ∈ V with u H ≤ n and v H ≤ n, and note that By (5.8) the first two terms can be estimated by L g u − v L 2 (R 3 ) .Concerning the last term we note that by (5.8) where in the last step we used ∇v L 2 ≤ n and Sobolev embedding with β 3 ∈ (3/4, 1).Similarly, by using (5.9), one can check ). (5.12) Step 3: (3.8) holds.By (5.11), (5.12), and the elementary estimate Recall that F 1 and F 2 are as in (5.10).For F 1 we have For F 2 , using that φ N (x) ≥ x − N for all x ≥ 0 gives where we used that φ N (x) = x − N for x ≥ N + 1. Combining the estimates for F 1 and F 2 , we obtain (5.13).It remains to apply Theorems 3.4 and 3.8.
Remark 5.6.One can also try to consider Dirichlet boundary conditions.When working with unweighted function spaces, this leads to serious difficulties as the noise needs to map into the right function spaces with boundary conditions (after applying the Helmholtz projection), which leads to strange assumptions.Moreover, the Helmholtz does not commute with the differential operators, which make the analysis more involved.Also the spaces are more involved since for V one needs to take the divergence free subspace of H 2 (O) ∩ H 1 0 (O) and for H the divergence free subspace of H 1 0 (O).In that way V * can be identified with the divergence free subspace of L 2 (O).Note that the divergence free subspace of C ∞ c (O) is not dense in V , and the dual (with respect to H) of the closure of the later space of test functions is not the divergence free subspace of L 2 (O) (cf.Example 2.2).
Remark 5.7.By estimating the F 1 term as for suitable δ ∈ (0, 1), and strengthening the stochastic parabolicity condition (5.7), one can also consider g with quadratic growth in the y-variable (see Subsection 5.4 for a related situation).
Remark 5.8.In Assumption 5.4, the condition b 2 ) for some δ > 0. We refer to the proof of Theorem 5.17 below for details.
(2) There exists ν > 0 such that a.e. on R (3) The mappings f : (4) There exist M, C ≥ 0 and η > 0 such that a.e. in R Condition ( 4) is technical, but can be seen as a direct translation of the coercivity condition (3.8).Simpler sufficient conditions will be give in Examples 5.13 and 5.15 below.Moreover, some simplification will also be discussed in Lemma 5.12.
In order to formulate (5.14) as

1).
Let A 0 : R + × Ω → L(V, V * ) and B 0 : R + × Ω → L(V, L 2 (U, H)) are given by We say that u is a solution to (5.14) if u is a solution to (3.3) with the above definitions.
In order to prove global well-posedness we check the conditions of Theorem 3.4.It is standard to check that Assumption 3.1(1)-( 2) are satisfied.To check (3) we only consider the local Lipschitz estimates, since the growth conditions can be checked in the same way.Note that for F 1 we have (by Sobolev embedding) (by Assumption 5.9) (by Hölder's inequality) (by Sobolev embedding).
First let d ≥ 3.In the first Sobolev embedding we choose r ∈ (1, ∞) such that The latter is equivalent to ρ 1 ≤ 4 d .If d = 1, then we can take r = 1, and this leads to , we can take r = 1 + δ for any δ > 0, and this leads to ρ 1 < 2. In all cases we can take 2β 1 = 1 + 1 ρ1+1 .To prove the estimates for f we argue similarly: (by Assumption 5.9) (by Hölder's inequality) (by Sobolev embedding).
As before we need Moreover, we can take 2β 3 = 1 + 1 ρ3+1 .Finally we note that the coercivity condition (3.8) coincides with Assumption 5.9(4).Therefore, Theorem 3.4 gives the following: Theorem 5.10.Suppose that Assumption 5.9 holds.Let u 0 ∈ L 0 F0 (Ω; L 2 (O)).Then (5.14) has a unique global solution s. and for every T > 0 there is a constant C T independent of u 0 such that Moreover, u depends continuously on the initial data u 0 in probability in the sense of Theorem 3.8 with H = L 2 (O) and V = H 1 0 (O).Remark 5.11.If Assumption 5.9 holds with η = 0, then a version of Theorem 5.10 still holds, but with (5.16) replaced by (3.11), (3.12), and (3.13).Indeed, instead one can apply Theorem 3.5.
In the next lemma we further simplify some of the terms appearing in Assumption 5.9(4) in special cases.Lemma 5.12.Suppose that Assumption 5.9 holds.Suppose that f and g only depends on (t, ω, y), and b only depends on (t, x, ω) and additionally div(b) = 0 in distributional sense.Then for al Therefore, Assumption 5.9(4) holds if there exist M, C ≥ 0 and η > 0 such that a.e. in R + × Ω for all y ∈ R, Proof.By extending u as zero we may assume that O is an open ball.In particular, this gives that O is a smooth domain.
(1): Let Since f is continuous, by the chain rule we obtain that for By the divergence theorem and the fact that u = 0 on ∂O, we obtain Therefore the stated result follows.
(2): We use a similar idea.Set Then the chain rule gives that Integrating by parts and arguing as before, we find that ((b n • ∇)u, g n (•, u)) L 2 (O) can be written as where in the last equality we used div b n = 0 that G n (•, u(x))| ∂O = 0 as u| ∂O = 0 and G(•, 0) = 0.For the final assertion note that by ( 1) and ( 2), Assumption 5.9(4) becomes By Assumption 5.9 for η small enough we can find θ > 0 such that The latter follows from (5.17).
Next we specialize to the setting of the generalized Burgers equation of [31,Example 5.1.8].It turns out that our setting leads to more flexibility under the mild restriction that the nonlinearities are locally Lipschitz with some polynomial growth estimate on the constants.Basically the natural restriction in our setting is given in (5.19) below which says that yf (y) ≤ M (1 + |y| 2 ) which is weaker than the usual one-sided Lipschitz condition used for the local monotonicity.Moreover, we can allow a gradient noise term.Further comparison can be found in Remark 5.14 below.For convenience we only consider coefficients which are independent of (t, ω, x), but in principle this is not needed.
Remark 5.14.For comparison let us note that the usual local monotonicity condition would require b = 0, and the more restrictive one-sided Lipschitz estimate Note that setting y = 0, the latter implies (5.19).Concerning the growth rate at infinity of order | • | ρi+1 in our setting and say the | • | ρ i +1 in the classical variational setting (see [31,Example 5.1.8]),we make a comparison in Table 1.In particular, in d = 2 the Burgers equation is included in our setting, but not in the classical variational framework.
The coercivity condition of Assumption 5.9(4) can be seen as a combined dissipativity condition on the nonlinearities f and g: better f gives less restrictions on g.Below, we give an explicit case in which it applies, where for simplicity we take f = 0 and b n = 0.
In case η = 0, one can use Remark 5.11 to obtain well-posedness.Here something special occurs in the case d = 1 and ρ ∈ (2, 3].In the latter case, (5.21) can be replaced by: there exists a C > 0 such that g(y) (5.22) The arguments below also apply if the term u − u 3 is replaced by a more general nonlinearity f (u) which behaves like u − u 3 .See Remark 5.20 for some comments.Table 1 in Remark 5.14 shows that (5.22) cannot be considered in the weak setting (i.e.H = L 2 , V = H 1 , and ρ 1 = 2) if d ≥ 2. In the current section we show that one can treat the stochastic Allen-Cahn equation for 2 ≤ d ≤ 4 by considering the strong setting instead (i.e.V = H 2 and H = H 1 ).In this setting local monotonicity does not holds, but fortunately local Lipschitz estimates are satisfied.Assumption 5.16.Let b j = (b j n ) n≥1 : R + ×Ω×T d ×R → 2 for j ∈ {1, . . ., d}, and g = (g n ) n≥1 : R + ×Ω×T d ×R → 2 be P ⊗B(T d )-and P ⊗B(T d )⊗B(R)-measurable maps, respectively.Assume that (1) Suppose that there exist ν ∈ (0, 2) such that a.e. on R (2) There exist M, δ > 0 such that for all j ∈ {1, . . ., d} a.s.for all t ∈ R + , (3) the mapping (x, y) → g(•, x, y) is C 1 (T d × R) a.e. on R + × Ω. (4) There exists C ≥ 0 such that a.e. on R + × Ω × T d and for all y, y ∈ R, (5) There exist C, η, θ > 0 such that for all v ∈ C ∞ (T d ) and a.e. on R + × Ω, Some remarks on Assumption 5.16 may be in order.(2) and the Sobolev embedding M .The quadratic growth assumption on g is optimal from a scaling point of view [6, Section 1]. ( 5) is equivalent to the coercivity condition (3.10).Note that the conditions (4) and ( 5) are compatible as the RHS of the estimate in (5) allows for quadratic growth of g.To check (5) it is also convenient to use that (5.23) The above follows by approximation by smooth functions, and integration by part arguments.As remarked above, Assumption5.16(5)allows g with quadratic growth.In special cases, one can even obtain explicit description of constants.For instance, if b = 0 and g is x-independent, then one can check that (5) holds if )) a.s.Moreover, for all T ∈ (0, ∞), there exists C T > 0 independent of u 0 such that Finally, u depends continuously on the initial data u 0 in probability in the sense of Theorem 3.8 with H = H 1 (T d ) and V = H 2 (T d ).
Remark 5.18.In case Assumption 5.16 (5) holds with η = 0, the above theorem still holds.However, the estimate (5.25) has to be replaced by the weaker bounds (3.11)-(3.13)with H = H 1 (T d ) and V = H 2 (T d ).The proof is the same as below, but one has to use Theorem 3.5 instead of Theorem 3.4.
As mentioned at the beginning of this subsection, the deterministic nonlinearity in (5.22) does not satisfy the classical local monotonicity condition for stochastic evolution equations, and therefore there are difficulties in applying the classical framework to obtain well-posedness for (5.22).
To estimate G, first observe that for all u, v ∈ H 2 (T d ) by Hölder's inequality and Sobolev embedding (using d ≤ 4), for k ∈ {0, 1}, Assumption 5.16( 5) and (5.28) imply that For the derivative term we can write By Assumption 5.16 (5), the first term can be estimated as before.For the second term Assumption 5.16 (5) gives Therefore, taking L 2 (T d )-norms and applying (5.28) we find that The growth estimate can be proved in a similar way, and thus Assumption 3.
Remark 5.21 (Kraichnan's noise).In the study of fluid flows transport noise (b n • ∇)u is typically used to model turbulence, see e.g.[14,15,25,33].In Kraichnan's theory, it is important to choose b as rough as possible.In this respect, Assumption 5.16(2) allows us to cover only regular Kraichnan's noise (see e.g.[17, Section 5] and the references therein).For the irregular case (e.g.b ∈ C ε ( 2 ) for ε > 0 small), Theorem 5.17 cannot be applied.However, by using L p -theory one can show that global well-posedness for (5.22) still holds, see [5].

5.5.
A stochastic quasi-linear second order equation.In this section we give an example of a quasi-linear SPDE to which our setting applies.Due to the quasi-linear structure we are forced to work with the strong setting V = H 2 and H = H 1 , since we need H → L ∞ .In case one would use L q -theory, then one can actually use a weaker setting, since the Sobolev embeddings theorems become better for q large.However, L q -theory is outside the scope of the current paper.
It is also possible to consider (t, ω) and space-dependent coefficients (a, b, f, g).In that case the conditions f (0) = 0 and g(0) = 0 can be weakened to an integrability condition.Note that a consequence of the Lipschitz estimate is that a In order to reformulate (5.29) as (3.3), we need some smoothness in the space H in order to deal with the quasi-linear terms.Therefore, let U Here we use (u, v) H 1 = (u, v) L 2 + (u , v ) L 2 and for the duality between V and V * we set u In order to prove local well-posedness we reformulate (5.29) in the form (3.3).Let and set A(u) = A 0 (u)u − F (u), and B(u) = B 0 (u)u + G(u).For local well-posedness it remains to check that Assumption 3.1 holds.
The following estimates will be used below: where for all ε > 0 the rest term satisfies H , where we used u ∞ ≤ m u H with m = Km.Taking ε ∈ (0, θ), Assumption 3.1(2) follows.
To check Assumption 3.1(3) let u, v ∈ V be such that u H , v H ≤ m and w ∈ V .Then by (5.30) The growth estimate is proved in the same way with C A0 m,T = C a Km.Similarly, The growth estimate can be proved analogously.Finally, to estimate F note that for all u, v ∈ V with u H , v H ≤ m where we used (5.30).The required growth estimate follows as well since F (0) = 0. Similarly, for G we can write The first term is clearly ≤ C u − v L 2 .The second term can be estimated as The growth estimate follows actually holds with m-independent constant.Indeed, since g (u) 2 ≤ C, we obtain From the above and Theorem 3.3 we obtain the following result.We do not know if the assumption on b can be avoided.Note that the condition on f holds for the important class of functions of the form f (x) = −c|x| h x + φ(x), where c, h > 0, and φ ∈ C 1 (R) is such that φ(x) ≤ C(1 + |x| h+1−δ ) and φ (x) ≤ C|x| h−δ for some C ≥ 0 and δ ∈ (0, h).Proof.By Theorem 3.4 it remains to check (3.8).Using that b is independent of u and taking u = v in (5.31), one can check that R can be estimated independently of m, and thus in the same way as before for η > 0 small enough Concerning the F -term we have   where we used M ≤ 2R + 1 and ln(2C) ≤ C. Since λ M = µ k , Chebychev's inequality implies (A.5) with τ R replaced by µ k .This completes the proof of (A.4).If η > 0, then we can add η on both sides of (A.1) and replace C by 2C.It remains to apply (A.4) to the pair (X + η, Y ).

. 8 )
also holds for some η > 0 and a slightly worse θ > 0. Moreover,(3.15)can even be weakened by replacing L 2 (U, H) by L(U, H) if one uses the weaker condition(3.14)instead.Note that (3.15) is usually assumed to hold in the classical framework (see(1.4)with α = 2).

4. 1 .
Proof of Theorem 3.3: local well-posedness and blow-up criteria.The proof of Theorem 3.3 consists of two parts.First we show local well-posedness via our recent result [3, Theorem 4.7].Proof of Theorem 3.3: local well-posedness.Let n ≥ 1 and set u

Example 5 . 13 .
Let d ≥ 1 and let O be a bounded C 1 -domain.Consider the problem(5.18)

2 2 ≤
C(1 + |y| 2 ) + C|y| 4 .5.4.Stochastic Allen-Cahn equation.The Allen-Cahn equation is one of the well-known reaction-diffusion equations of mathematical physics, and it is used to describe phase transition processes.Here we consider the following stochastic Allen-Cahn equation with transport noise on T d : − ∆u dt = (u − u 3 )dt + n≥1 (b n • ∇)u + g n (•, u) dw n t , on T d , u(0) = u 0 on T d .

30 )
Note that by Sobolev embeddingV → C 1 b (R) and H → C b (R).Assumption 3.1(1) is simple to check.To check condition Assumption 3.1(2) note that for all u ∈ H with u H ≤ m and v ∈ V ,(5.31)